Burling graphs in graphs with large chromatic number

TL;DR

This work identifies Burling graphs as canonical witnesses of large chromatic number within broad geometric and hereditary graph classes that exclude induced subdivisions of a fixed graph. By developing the Burling-control framework—packing arguments built from multicovers, subdivision-exclusion, and homomorphisms from Burling graphs—the authors show that unbounded χ in string, region-intersection, and finite-asymptotic-dimension classes arises only from large cliques or Burling graphs, not from arbitrary substructures. They prove that Burling graphs constitute a minimal hereditary class with unbounded χ and deduce polynomial-time algorithms for approximate coloring by searching for a large clique or a Burling copy, with extensions to region and string graphs and to classes with finite asymptotic dimension. The results generalize and unify several prior χ-boundedness questions for geometric intersection graphs, yielding new algorithmic consequences and suggesting Burling graphs as central witnesses in the study of chromatic number in restricted graph classes.

Abstract

A graph class is $χ$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erdős conjectured that intersection graphs of straight-line segments in the plane are $χ$-bounded, but this was disproved by Pawlik et al. (2014), who showed another way to force large chromatic number in this class -- by triangle-free graphs $B_k$ with $χ(B_k)=k$ constructed by Burling (1965). This also disproved the celebrated conjecture of Scott (1997) that classes of graphs excluding induced subdivisions of a fixed graph are $χ$-bounded. We prove that in broad classes of graphs excluding induced subdivisions of a fixed graph, including the increasingly more general classes of segment intersection graphs, string graphs, region intersection graphs, and hereditary classes of graphs with finite asymptotic dimension, large chromatic number can be forced only by large cliques or large graphs $B_k$. One corollary is that the hereditary closure of $\{B_k\colon k\geq 1\}$ forms a minimal hereditary graph class with unbounded chromatic number -- the second known graph class with this property after the class of complete graphs. Another corollary is that the decision variant of approximate coloring in the aforementioned graph classes can be solved in polynomial time by exhaustively searching for a sufficiently large clique or copy of $B_k$. We also discuss how our results along with some results of Chudnovsky, Scott, and Seymour on the existence of colorings can be turned into polynomial-time algorithms for the search variant of approximate coloring in string graphs (with intersection model in the input) and other aforementioned graph classes. Such an algorithm has not yet been known for any graph class that is not $χ$-bounded.

Figures (3)

• Figure 1: Schematic depiction of the inductive construction of $B_{k+1}$ (\ref{['fig:burling-step']}) and $B'_{k+1}$ (\ref{['fig:new-step']}). The vertices in $V_A(B_{k+1})$ and $V_A(B'_{k+1})$ are depicted in red. Purple vertices are the ones in $V_A(B_k)$ that are not vertices of $B_{k+1}$.
• Figure 2: A multicover with a skewer
• Figure 3: Schematic depiction of the operations on marked graphs. Vertices in $V_A(\mathop{\mathrm{\mathsf{Sub}}}\nolimits(G,H))$ and $V_A(\mathop{\mathrm{\mathsf{Ext}}}\nolimits(G,H))$ are depicted in red. Vertices in $V_A(\mathop{\mathrm{\mathsf{Sub}}}\nolimits^*(G,H))$ are the red and the purple ones, where the latter are not vertices of $\mathop{\mathrm{\mathsf{Sub}}}\nolimits(G,H)$.

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6: Chudnovsky, Scott, and Seymour CSS21
• Theorem 1.7
• Proposition 2.1: Burling Bur65
• Lemma 2.1
• Proposition 2.2: Sinden Sin66